What are Vieta's formulas?

Vieta's formulas express relationships between the coefficients of a polynomial and sums and products of its roots, without needing to find the roots explicitly.

Can this find the actual roots?

This calculator shows the sum and product of roots via Vieta's formulas. For exact roots, factoring or numerical methods are typically needed for cubics and higher.

What does the discriminant indicate?

A positive discriminant means three distinct real roots. Zero means at least one repeated root. Negative means one real root and two complex conjugate roots.

Polynomial Roots Calculator

Analyze a cubic polynomial ax³ + bx² + cx + d. Shows the sum and product of roots via Vieta's formulas.

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Coefficient a (x³)

Coefficient b (x²)

Coefficient c (x)

Coefficient d (constant)

Sum Roots

6.0000

Sum Products Pairs11.0000

Discriminant4.0000

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Polynomial Roots via Vieta's Formulas

For a Cubic ax³ + bx² + cx + d = 0

Vieta's formulas relate the roots (r₁, r₂, r₃) to the coefficients:

r₁ + r₂ + r₃ = -b/a (sum of roots)

r₁r₂ + r₁r₃ + r₂r₃ = c/a (sum of pairwise products)

r₁r₂r₃ = -d/a (product of roots)

Discriminant

The discriminant tells you about the nature of the roots:

Δ > 0: three distinct real roots

Δ = 0: a repeated root

Δ < 0: one real root and two complex conjugate roots

Example Calculation

Analyze x³ - 6x² + 11x - 6 = 0 (roots are 1, 2, 3).

01Sum of roots = -(-6)/1 = 6 (= 1+2+3)
02Sum of pairwise products = 11/1 = 11 (= 1×2+1×3+2×3)
03Product of roots = -(-6)/1 = 6 (= 1×2×3)

Frequently Asked Questions

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Polynomial Roots Calculator

Polynomial Roots via Vieta's Formulas

For a Cubic ax³ + bx² + cx + d = 0

Discriminant

Example Calculation

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